This model is a regularization of Euler equations describing the flow of an incompressible fluid on spatial scales larger than the length scale parameter \(\alpha .\)  It has been introduced in the context of averaged fluid models, see [1]}, [2]}. In the literature, \(\mathbf {v}\)  and \(\mathbf {u}\)  are called the filtered and unfiltered velocities, respectively. Actually, \(\mathbf {u}\)  corresponds to the velocity field of the fluid. Notice that \((\nabla \mathbf {v})^{\top }\)  denotes the transpose of the Jacobi matrix for \(\mathbf {v}\) , namely
\((\nabla \mathbf {v})^{\top }=\left(\frac{\partial \mathbf {v}_j}{\partial x_i}\right)_{1\leqslant i,j\leqslant 2}\) 
